1 Find the area of the part of the surface 2z = x^2 that lies directly above the triangle in the xy-plane with the vertices at (0,0),(1,0) and (1,1).
2 Find the volume of the region in the first octant that is bounded by the hyperbolic cylinders xy = 1, xy = 9, xz = 4, yz = 1, and yz = 16. Use the transformation u = xy, v =

Let f(z) be holomorphic in |z|less than R with Taylor expansion f(z)=sum(a_nz^n) and set
I_2(r)=1/2pi(integral from 0 to 2pi of|f(re^itheta)|^2 d(theta), where 0<=r<R.
Show that
a) I_2(r)=sum(n=0 to 00)|a_n|^2r^2n
b) I_2(r) is increasing.
c) |f(0)|^2<=I_2(r)<=M(r)^2, with M(r)=sup_|z||f(z)|

6. Determine the divergence or convergence of the given improper integral. Evaluate the integral if it converges.
integral to the power of 4 sub 3 1 / (square root x - 3) dx
7. Use integration by parts to evaluate the given integral.
integral x sec^2 x dx

Explain why the derivative function of the function g(x) = x is equal to 1 on the interval (0,&#8734;), equal to 1 on the interval
(-&#8734;,0), and undefined at 0. [Hint. Sketch the graph of g.]
Consider the region R bounded by the curve xy =3 and the lines x I and x =4. Set up the integrals (do not evaluate) that give the

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes?
keywords: integration, integrates, integrals, integrating, double, triple, m

1. Given f &#8242;(x) = ex + (4/3)x^(-2/3) , find f (x) if f(1) = e.
2. Given f (x) = x2 + x +1
(a) Approximate the area between the curve of f and the x-axis on the interval [0,2] using 4 rectangles
and right point sums.
(b) Find the EXACT area between the curve of f and the x-axis on the interval [0,2] by using area a

Let S be the closed surface of the paraboloid of revolution z = ±(4 &#8722; x2 &#8722; y2 ) where &#8722;2 x, y +2. Evaluate the following surface integral directly and then by using the divergence theorem; where R is the position vector to a point on the surface and is the outward pointing normal at that point.
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